Wedgie/Archived version: Difference between revisions

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<math>\left(E_{19}\wedge E_{31}\right)\left(2,3\right) = E_{19}\left(2\right)E_{31}\left(3\right) - E_{19}\left(3\right)E_{31}\left(2\right) = 19*49 - 31*30 = 1.</math>

We may continue in this way to consider (2,5), (2,7), (3,5), (3,7), and (5,7), and writing them in this alphabetical order yields <math>\bitval{1 & 4 & 10 & 4 & 13 & 12}</math>. Here, the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank two temperament]]s such as [[meantone]], trivals to [[rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, {{nowrap|E19 ∧ E31}} is the same object we were calling {{nowrap|"meantone(''u'', ''v'')"}} which gives us complexity measurements for meantone.

We may continue in this way to consider (2, 5), (2, 7), (3, 5), (3, 7), and (5, 7), and writing them in this alphabetical order yields <math>\bitval{1 & 4 & 10 & 4 & 13 & 12}</math>. Here, the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank two temperament]]s such as [[meantone]], trivals to [[rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, {{nowrap|E19 ∧ E31}} is the same object we were calling {{nowrap|"meantone(''u'', ''v'')"}} which gives us complexity measurements for meantone.

This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the {{w|GCD}} of all of the coordinates is 1. An n-map with these properties we may call ''reduced'', and reduced n-vals can be used to give unique names to [[regular temperament]]s.

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 <math>\left(E_{19}\wedge E_{31}\right)\left(2,3\right) = E_{19}\left(2\right)E_{31}\left(3\right) - E_{19}\left(3\right)E_{31}\left(2\right) = 19*49 - 31*30 = 1.</math>
-We may continue in this way to consider (2,5), (2,7), (3,5), (3,7), and (5,7), and writing them in this alphabetical order yields <math>\bitval{1 & 4 & 10 & 4 & 13 & 12}</math>. Here, the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank two temperament]]s such as [[meantone]], trivals to [[rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, {{nowrap|E<sub>19</sub> ∧ E<sub>31</sub>}} is the same object we were calling {{nowrap|"meantone(''u'', ''v'')"}} which gives us complexity measurements for meantone.
+We may continue in this way to consider (2,&nbsp;5), (2,&nbsp;7), (3,&nbsp;5), (3,&nbsp;7), and (5,&nbsp;7), and writing them in this alphabetical order yields <math>\bitval{1 & 4 & 10 & 4 & 13 & 12}</math>. Here, the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank two temperament]]s such as [[meantone]], trivals to [[rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, {{nowrap|E<sub>19</sub> ∧ E<sub>31</sub>}} is the same object we were calling {{nowrap|"meantone(''u'', ''v'')"}} which gives us complexity measurements for meantone.
 This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the {{w|GCD}} of all of the coordinates is 1. An n-map with these properties we may call ''reduced'', and reduced n-vals can be used to give unique names to [[regular temperament]]s.